What does $\mathbb{Z}[X]$ for a polynom mean?

Question

I have a proof saying that a Polynom $p \in \mathbb{Z}[X_1,...,X_m]$

I'm a bit confused of this notation because neither X nor the m is explains somewhere.

Does somebody of you know the notation?

Answer 1

If it helps, think of $\mathbb{Z}[X_1,X_2,X_3] $ as $\mathbb{Z}[x,y,z]$ and then your favorite polynomials (in three variables) from multivariable calculus are in this ring. For instance, for $z^2=x^2+y^2$, you have $z^2-x^2-y^2 \in \mathbb{Z}[x,y,z]$. But for an arbitrary polynomial in $m$ variables it is better to write $X_1,...,X_m$ than choosing $m$ symbols for each variable. Once the notation has been established, one writes $X$ in place of $X_1,...,X_m$. So $\mathbb{Z}[X_1,...,X_m]=\mathbb{Z}[X]$ and $f(X_1,...,X_m)\in \mathbb{Z}[X_1,...,X_m]$ is simply written as $f(X) \in \mathbb{Z}[X]$.

Answer 2

This means that $p$ is a polynomial with integer coefficients in $m$ variables. An example would be $$ p( X_1 , X_2 , \ldots , X_m) = X_1 + X_2 + \cdots + X_m. $$